2.2.3. Variable interest rate

It should be noted that the basic compound interest formula involves permanent interest rate throughout the interest period. However, when providing a long-term loan, they often use time-varying, but pre-fixed rates for each period. compound interest. In case of use variables interest rates, the accrual formula is as follows:

where ik– successive values ​​of interest rates in time;

nk– the duration of the periods during which the corresponding rates are used.

Example. The company received a loan from a bank in the amount of $ 100,000 for a period of 5 years. The interest rate on the loan is set at 10% for the 1st year, for the 2nd year there is a surcharge on the interest rate of 1.5%, for subsequent years 1% Determine the amount of debt due at the end of the loan term.

Solution:

We use the formula for variable interest rates:

FV=PV (1 + i 1)n 1 (1 + i 2)n 2 … (1 + ik)nk =

100"000 (1 + 0,1) (1 + 0,115) (1 + 0,125) 3 =

174"$632.51

Thus, the amount due at the end of the loan term will be $174,632.51, of which $100,000 is directly owed and $74,632.51 is interest on the debt.

2.2.4. Continuous interest calculation

All situations that we have considered so far have been discrete interest, since they are calculated over fixed periods of time (year, quarter, month, day, hour). However, in practice, there are often cases when interest accrues continuously, for an arbitrarily short period of time. If interest were accrued daily, then the annual coefficient (multiplier) of accumulation would look like this:

kn = (1 + j / m)m = (1 + j / 365) 365

But since interest is accrued continuously, then m tends to infinity, and the accumulation coefficient (multiplier) tends to ej:

where e≈ 2.718281 is called the Euler number and is one of the most important constants in mathematical analysis.

From here, we can write the formula for the accumulated amount for n years:

FV = PV e j n = P e δ n

The continuous interest rate is called force of interest and are symbolized δ , in contrast to the discrete interest rate ( j).

Example. A loan in the amount of 100 thousand dollars was received for a period of 3 years at 8% per annum. Determine the amount to be repaid at the end of the loan term, if interest will accrue:

a) once a year;

b) daily;

c) continuously.

Solution:

We use the formulas for discrete and continuous percentages:

accrual once a year

FV\u003d 100 "000 (1 + 0.08) 3 \u003d 125" 971.2 dollars;

daily interest calculation

FV= 100 "000 (1 + 0.08 / 365) 365 3 = 127" $121.6

continuous interest

FV\u003d 100 "000 e 0.08 3 \u003d 127" 124.9 dollars.

Graphically, the change in the accrued amount depending on the frequency of accrual has the following form:

With discrete accrual, each "step" characterizes the increase in the principal amount of the debt as a result of the next interest accrual. Please note that the height of the "steps" is increasing all the time.

Within one year, one "step" on the left graph corresponds to two "steps" on the middle graph of a smaller size, but in total they exceed the height of the "step" of a single accrual. Even more rapidly is the accumulation with the continuous calculation of interest, as shown by the graph on the right.

Thus, depending on the frequency of interest accrual, the accumulation of the initial amount is carried out at different rates, and the maximum possible accumulation is carried out with an infinite splitting of the annual interval.

Continuous interest calculation is used in the analysis of complex financial problems, such as the rationale and selection of investment decisions. When evaluating the work of a financial institution where payments for a period are received repeatedly, it is reasonable to assume that the accrued amount changes continuously over time and apply continuous interest calculation.

2.2.5. Loan term and interest rate determination

Just like for simple interest, for compound interest it is necessary to have formulas that allow you to determine the missing parameters of a financial transaction:

loan term:

n = / = / ;

compound interest rate:

Thus, a three-fold increase in a deposit over three years is equivalent to an annual interest rate of 44.3%, so placing money at 46% per annum will be more profitable.

2.3. Equivalence of rates and replacement of payments

2.3.1. Interest rate equivalence

Quite often in practice, a situation arises when it is necessary to compare the terms of various financial transactions and commercial transactions in terms of profitability. The conditions of financial and commercial transactions can be very diverse and directly incomparable. For comparison of alternative options, the rates used in the terms of contracts are brought to a uniform rate.

Equivalent interest rate - this is the rate that for the financial transaction in question will give exactly the same monetary result (accumulated amount) as the rate used in this transaction.

The classic example of equivalence is nominal and effective rate percent:

i = (1 + j / m)m - 1.

j = m[(1 + i) 1 / m - 1].

The effective rate measures the relative income that can be received in the whole year, i.e. it is completely indifferent whether to apply the rate j when calculating interest m once a year or annual rate i, – both rates are financially equivalent.

Therefore, it does not matter at all which of the given rates is indicated in the financial terms, since using them gives the same accrued amount. In the United States, the nominal rate is used in practical calculations, while in European countries they prefer the effective interest rate.

If two nominal rates determine the same effective interest rate, they are called equivalent.

Example. What would be the equivalent nominal interest rates with semi-annual interest and monthly interest if the corresponding effective rate should be equal to 25%?

Solution:

We find the nominal rate for the semi-annual interest calculation:

j = m[(1 + i) 1 / m - 1] = 2[(1 + 0,25) 1/2 - 1] = 0,23607.

We find the nominal rate for the monthly interest calculation:

j = m[(1 + i) 1 / m - 1] = 4[(1 + 0,25) 1/12 - 1] = 0,22523.

Thus, the nominal rates of 23.61% with semi-annual interest and 22.52% with monthly interest are equivalent.

When deriving equalities linking equivalent rates, the accumulation multipliers are equated to each other, which makes it possible to use the formulas for the equivalence of simple and complex rates:

simple interest rate:

i = [(1 + j / m)m n - 1] / n;

compound interest rate:

Example. It is supposed to place the capital for 4 years either at a compound interest rate of 20% per annum with semi-annual interest, or at a simple interest rate of 26% per annum. Find the best option.

Solution:

Find the equivalent simple rate for the compound interest rate:

i = [(1 + j / m)m n - 1] / n = [(1 + 0,2 / 2) 2 4 - 1] / 4 = 0,2859.

Thus, the simple interest rate equivalent to the compound rate under the first option is 28.59% per annum, which is higher than the proposed simple rate of 26% per annum under the second option, therefore, it is more profitable to place capital under the first option, i.e. at 20% per annum with semi-annual interest.

A discrete interest rate is a rate at which interest is charged for predetermined, or specified, periods. If you reduce the interest calculation period to an infinitesimal value (the period for which accruals will be made tends to zero, and the number of interest accruals tends to infinity), then interest will be accrued continuously. In this case, the interest rate is called continuous rate or force of growth .

In theoretical studies and in practice, when payments are made repeatedly, it is convenient to use the continuous method of interest calculation. The transition to the limit can be carried out in the same way as it was done in paragraph 2.2 when deriving formula (2.12) or in the following way.

The continuous rate can be fixed or variable. Consider the case when the continuous interest rate is different at different times.

Let а(t) be a function describing the dependence of the continuous rate (growth force) on time t. The increment of capital S(t) at the moment t for the time interval Δt is equal to:

S(t + Δt) – S(t) = a(t) Δt S(t)

Then, we have:

When Δt →0 we get that the rate of change of capital is proportional to capital. Then, the payment amount (capital) S(t) satisfies the first-order linear homogeneous differential equation:

, (2.28)

– rate of change of payment (rate of change of capital);

S(t) - payment amount (capital);

a(t) - continuous accrual percentage or growth force.

In another form, the equation will be written:

dS = a(t) S dt, (2.29)

i.e., the payment increment is proportional to the payment S itself and the time increment dt. The coefficient of proportionality a(t) is the force of growth or the percentage of accrual.

There is another way to write the differential equation:

, (2.30)

i.e., the relative increment of the payment amount dS/S is proportional to the increment of time dt. Moreover, as before, a(t) is determined by the percentage of accrual and, in the general case, may depend on time. All three capital equations (2.28), (2.29), (2.30) are equivalent.

Consider some of the simplest properties of capital, described by the differential equation (2.28)-(2.30). If the function a(t)>0 is positive, then with a positive capital S>0, the derivative of the capital dS/dt >0 is also positive and, consequently, the capital S(t) grows. In this case a(t) is called continuous interest accrual or force of growth .

Otherwise, if the function a(t)<0 отрицательна, то при положительном капитале S>0 capital derivative dS/dt<0 отрицательна и, следовательно, капитал S(t) убывает. В этом случае абсолютная величина |a(t)| называется continuous discount .

The solution of a linear differential equation is well known. Indeed, equation (2.30) is an equation with separable variables and can be integrated:

Calculating the integral, we get:

,

where - indefinite integral of a(t),

C 1 is an arbitrary constant.

Hence, we have:

Finally, the general solution of the differential equation can be written as:

, (2.31)

where is a new arbitrary constant.

To define an arbitrary constant FROM you need to know the capital at least at one point in time. If it is known that at time t=t 0 the capital is equal to S = S 0 (i.e. S(t 0)=S 0), then an arbitrary constant FROM is easily determined from (2.31):

,

Substituting the result obtained into (2.31), we have:

.

Using the classical formula for the connection of a definite and indefinite integral (the Newton-Leibniz formula):

,

we obtain the solution of the differential equation with the initial conditions S(t 0)=S 0 in the form:

Often time can be measured from the initial moment, then t 0 =0 and the solution of the linear differential equation is written as:

, (2.32)

S(0) is the initial amount at time 0;

S(t) is the payment amount at time t.

Obviously, the above formulas for a(t)>0 correspond to the calculation of lending, and for a(t)<0 – расчету дисконтирования.

If the force of growth is constant throughout the considered time interval, i.e. a(t)= r, then for the final payment at time t we have:

. (2.33)

Obviously, this formula coincides with the formula (2.12) obtained earlier by passing to the limit.

Let's consider some examples of using these formulas.

Example 28.

Loan of 200 thousand rubles. given for 2.5 years at a rate of 20% per annum with a quarterly accrual. Find the amount of the final payment. Calculation to be made on discrete and continuous percent.

Solution.

The amount of the final payment satisfies the differential equation , where r=20%=0.2 in accordance with the percentage of annual accrual and the time t is measured in years. The solution of the linear equation is known:

.

Then the final payment is:

Thousand rub.

Calculation for the discrete case by formulas (2.11) gives:

Thousand rub.

It can be seen that with multiple accruals of small interest, the results of calculating the amounts of the final payment are close.

Consider now an example of calculating discounting in the continuous case.

Example 29.

Promissory note for 3 million rubles. with an annual discount rate of 10% and discounted twice a year issued for 2 years. Find the original amount to be lent against this bill. Calculation to be made on discrete and continuous percent.

Solution.

The payment amount borrowed against a bill of exchange satisfies a linear differential equation, the solution of which is known:

.

The calculation of the amount borrowed against a bill using discrete formulas (2.24) gives similar results:

mln rub.

Thus, theoretical and practical calculations using continuous formulas give results close to the results of calculating using discrete formulas, if the number of accruals is large, and the percentage of accrual is small.

In practically financial and credit operations, a continuous increase, i.e. build-up over infinitesimal periods of time is used extremely rarely. Continuous accumulation is of much greater importance in the analysis of complex financial problems, for example, in substantiating and choosing investment decisions, in financial design.

With a continuous increase in interest, a special type of interest rate is used - the force of growth.

Strength of Growth characterizes the relative increase in the accumulated amount over an infinitely small period of time. It can be constant or change over time.

In order to distinguish a continuous rate from a discrete rate, we denote the growth rate as δ . Then the accumulated amount at a continuous rate will be:

Discrete and continuous accrual rates are functionally dependent. From the equality of the multipliers of the increase

follows: ,

.

Example: The amount on which continuous interest is charged is 2 million rubles, the growth rate is 10%, the term is 5 years. Determine the accumulated amount.

A continuous increase at a rate = 10% is equivalent to an increase over the same period of discrete compound interest at an annual rate:

As a result, we get:

Discount formula:

.

The discount factor is .

Example: Determine the current value of the payment if the accrued value is 5,000 thousand rubles. subject to discounting by growth force of 12%. The payment term is 5 years.

When using a discrete nominal rate, the accumulated amount is determined by the formula:

When switching to continuous percentages, we get:

Accrual multiplier for continuous capitalization of interest.

Denoting the force of growth through, we get:

because discrete and continuous rates are functionally related to each other, then we can write the equality of the multipliers of the accumulation

For the initial capital 500 thousand rubles. accrued compound interest - 8% per annum for 4 years. Determine the accumulated amount if interest is accrued continuously.

Discounting based on continuous interest rates

In formula (4.21) one can determine the modern value

The continuous interest rate used in discounting is called the discount power. It is equal to the strength of growth, i.e. used to discount the discount force or the growth force lead to the same result.

Define the present value of the payment, assuming that discounting is done at a growth rate of 12% and at a discrete compound discount rate of the same size.

Variable strength of growth

With the help of this characteristic, the processes of increasing money amounts with a changing interest rate are modeled. If the force of growth is described by some continuous function of time, then the formulas are valid.

For the accumulated amount:

Modern value:

1) Let the growth force change discretely and take the following values: in time intervals, then at the end of the loan term, the accumulated amount will be:

If the accumulation period is n, and the average growth value is: , then

Determine the accrual multiplier for continuous interest accrual for 5 years. If the strength of growth changes discretely and corresponds to: 1 year - 7%, 2 and 3 - 8%, the last 2 years - 10%.

2) The growth force continuously changes in time and is described by the equation:

where is the initial force of growth (at)

a - annual increase or decrease.

Calculate the degree of the multiplier of the increase:

Initial value growth force of 8%, the interest rate is continuous and linear.

Growth for the year -2%, accumulation period - 5 years. Find the growth factor.

3) The strength of growth changes exponentially, then

In practical financial and credit operations, a continuous increase, i.e. build-up over infinitesimal periods of time is used extremely rarely. Of much greater importance is continuous accumulation in the analysis of complex financial problems, for example, in the justification and selection of investment decisions.

The accrued amount at discrete interest is determined by the formula

S=P(1+j/m) mn ,

where j is the nominal interest rate, and m is the number of interest periods per year.

The more m, the shorter the time intervals between the moments of interest calculation. Increasing the frequency of interest calculation ( m) at a fixed value of the nominal interest rate j leads to an increase in the accrual multiplier, which, with the continuous calculation of interest ( m) reaches its limit value

It is known that

where e is the base of natural logarithms.

Using this limit in expression (2.5), we finally obtain that the accrued amount at the rate j is equal to

S=Pe jn .

The continuous rate of interest is called the force of growth and is denoted by the symbol  . Then

S=Pe  n . (2.6)

Strength of Growth is the nominal interest rate at m.

The accrual law for continuous interest calculation (2.6) coincides in form with (2.2) with the difference that in (2.2) the time changes discretely with a step 1/ m, and in (2.6) it is continuous.

It is easy to show that discrete and continuous accrual rates are in a functional relationship. From the equality of accrual multipliers, we can obtain a formula for the equivalent transition from one rate to another:

(1+i) n =e  n ,

from where follows:

=ln(1+ i), i=e  -1.

Example 20 . The amount on which continuous interest is charged for 5 years is 2000 den. units, growth force 10%. The accumulated amount will be S=2000 e 0.1 5 \u003d 2000 1.6487 \u003d 3297.44 den. units

A continuous increase at a rate of 10% is equivalent to an increase over the same period of compound discrete interest at an annual rate i. We find:

i=e 0,1 -1=1,10517-1=0,10517.

As a result, we get S\u003d 2000 (1 + 0.10517) 5 \u003d 3297.44 den. units

Discounting based on the strength of growth is carried out according to the formula

P=Se -  n

Example 21. Let's determine the present value of the payment from example 17, provided that the discounting is based on the growth rate of 15%.

Solution. The amount received for the debt (modern value) is equal to

P=5000 e-0.15 5 \u003d 5000 0.472366 \u003d 2361.83 den. units

When applying a discrete complex discount rate of the same size, we obtained the value (see example 17) P=2218.53 den. units

2.5. Calculation of the loan term and interest rates

In a number of practical tasks, the initial (P) and final (S) amounts are specified by the contract, and it is required to determine either the payment term or the interest rate, which in this case can serve as a measure of comparison with market indicators and a characteristic of the profitability of the operation for the lender. These values ​​are easy to find from the original accrual and discount formulas (for simple interest, these problems are discussed in paragraph 1.8.).

Loan term. Consider the calculation problem n for different terms of accrual of interest and discounting.

i from the original growth formula (2.1) it follows that

,

where the logarithm can be taken in any base, since it is present in both the numerator and the denominator.

j m

.

d f m

;

.

With an increase in constant growth force, based on formula (2.6), we obtain:

.

Example 22. For what period in years is the amount equal to 75 thousand den. units, will reach 200 thousand den. units when accruing interest at a compound rate of 12% once a year and quarterly?

Solution. According to the formulas for calculating the term when accruing at complex accrual rates, we get:

n=(log(200/75)/log(1+0.12))=3.578 years;

n=(log(200/75)/(4 log(1+0.12/4))=3.429 years;

Calculation of interest rates. From the same initial formulas as above, we obtain formulas for calculating rates under various conditions for accruing interest and discounting.

When accruing at a compound annual rate i from the original growth formula (2.1) it follows that

i=(S/P) 1/ n –1=
.

When accruing at the nominal rate of interest m once a year from formula (2.2) we get:

j=m((S/P) 1/ mn –1)=
.

When discounted at a compound annual discount rate d and at the nominal discount rate f m once a year from formulas (2.3) and (2.4), respectively, we obtain:

d =1– (P/S) 1/ n =
;

f = m(1– (P/S) 1/ mn =
.

With an increase in constant force of growth, based on formula (2.6), we obtain:

.

Example 23. Savings certificate bought for 100 thousand den. units, its redemption amount is 160 thousand den. units, term 2.5 years. What is the rate of return on an investment in the form of an annual compound interest rate?

Solution. Using the resulting formula for the annual rate i, we get: i=(160/100) 1/2.5 –1=1.2068–1=0.20684, i.e. 20.684%.

Example 24. The maturity of the promissory note is 2 years. The discount at its accounting was 30%. What compound annual discount rate does this discount correspond to?

Solution. According to the task P/S=0.7. Then d=1–
=0.16334, i.e. 16.334%.